author wenzelm Thu, 08 Aug 2002 23:47:41 +0200 changeset 13482 2bb7200a99cf parent 13481 796a5b766c2e child 13483 0e6adce08fb0
converted;
 src/HOL/Integ/Equiv.ML file | annotate | diff | comparison | revisions src/HOL/Integ/Equiv.thy file | annotate | diff | comparison | revisions
--- a/src/HOL/Integ/Equiv.ML	Thu Aug 08 23:46:51 2002 +0200
+++ b/src/HOL/Integ/Equiv.ML	Thu Aug 08 23:47:41 2002 +0200
@@ -1,262 +1,38 @@
-(*  Title:      Equiv.ML
-    ID:         $Id$
-    Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
-    Copyright   1996  University of Cambridge

-Equivalence relations in HOL Set Theory
-*)
-
-(*** Suppes, Theorem 70: r is an equiv relation iff r^-1 O r = r ***)
-
-(** first half: equiv A r ==> r^-1 O r = r **)
-
-Goalw [trans_def,sym_def,converse_def]
-     "[| sym(r); trans(r) |] ==> r^-1 O r <= r";
-by (Blast_tac 1);
-qed "sym_trans_comp_subset";
-
-Goalw [refl_def] "refl A r ==> r <= r^-1 O r";
-by (Blast_tac 1);
-qed "refl_comp_subset";
-
-Goalw [equiv_def] "equiv A r ==> r^-1 O r = r";
-by (Clarify_tac 1);
-by (rtac equalityI 1);
-by (REPEAT (ares_tac [sym_trans_comp_subset, refl_comp_subset] 1));
-qed "equiv_comp_eq";
-
-(*second half*)
-Goalw [equiv_def,refl_def,sym_def,trans_def]
-     "[| r^-1 O r = r;  Domain(r) = A |] ==> equiv A r";
-by (etac equalityE 1);
-by (subgoal_tac "ALL x y. (x,y) : r --> (y,x) : r" 1);
-by (ALLGOALS Fast_tac);
-qed "comp_equivI";
-
-(** Equivalence classes **)
-
-(*Lemma for the next result*)
-Goalw [equiv_def,trans_def,sym_def]
-     "[| equiv A r;  (a,b): r |] ==> r{a} <= r{b}";
-by (Blast_tac 1);
-qed "equiv_class_subset";
-
-Goal "[| equiv A r;  (a,b): r |] ==> r{a} = r{b}";
-by (REPEAT (ares_tac [equalityI, equiv_class_subset] 1));
-by (rewrite_goals_tac [equiv_def,sym_def]);
-by (Blast_tac 1);
-qed "equiv_class_eq";
-
-Goalw [equiv_def,refl_def] "[| equiv A r;  a: A |] ==> a: r{a}";
-by (Blast_tac 1);
-qed "equiv_class_self";
-
-(*Lemma for the next result*)
-Goalw [equiv_def,refl_def]
-    "[| equiv A r;  r{b} <= r{a};  b: A |] ==> (a,b): r";
-by (Blast_tac 1);
-qed "subset_equiv_class";
-
-Goal "[| r{a} = r{b};  equiv A r;  b: A |] ==> (a,b): r";
-by (REPEAT (ares_tac [equalityD2, subset_equiv_class] 1));
-qed "eq_equiv_class";
-
-(*thus r{a} = r{b} as well*)
-Goalw [equiv_def,trans_def,sym_def]
-     "[| equiv A r;  x: (r{a} Int r{b}) |] ==> (a,b): r";
-by (Blast_tac 1);
-qed "equiv_class_nondisjoint";
-
-Goalw [equiv_def,refl_def] "equiv A r ==> r <= A <*> A";
-by (Blast_tac 1);
-qed "equiv_type";
-
-Goal "equiv A r ==> ((x,y): r) = (r{x} = r{y} & x:A & y:A)";
-qed "equiv_class_eq_iff";
-
-Goal "[| equiv A r;  x: A;  y: A |] ==> (r{x} = r{y}) = ((x,y): r)";
-qed "eq_equiv_class_iff";
-
-(*** Quotients ***)
-
-(** Introduction/elimination rules -- needed? **)
-
-Goalw [quotient_def] "x:A ==> r{x}: A//r";
-by (Blast_tac 1);
-qed "quotientI";
-
-val [major,minor] = Goalw [quotient_def]
-    "[| X:(A//r);  !!x. [| X = r{x};  x:A |] ==> P |]  \
-\    ==> P";
-by (resolve_tac [major RS UN_E] 1);
-by (rtac minor 1);
-by (assume_tac 2);
-by (Fast_tac 1);   (*Blast_tac FAILS to prove it*)
-qed "quotientE";
-
-Goalw [equiv_def,refl_def,quotient_def]
-     "equiv A r ==> Union(A//r) = A";
-by (Blast_tac 1);
-qed "Union_quotient";
-
-Goalw [quotient_def]
-     "[| equiv A r;  X: A//r;  Y: A//r |] ==> X=Y | (X Int Y = {})";
-by (Clarify_tac 1);
-by (rtac equiv_class_eq 1);
-by (assume_tac 1);
-by (rewrite_goals_tac [equiv_def,trans_def,sym_def]);
-by (Blast_tac 1);
-qed "quotient_disj";
-
-
-(**** Defining unary operations upon equivalence classes ****)
-
-(* theorem needed to prove UN_equiv_class *)
-Goal "[| a:A;  ALL y:A. b(y)=c |] ==> (UN y:A. b(y))=c";
-by Auto_tac;
-qed "UN_constant_eq";
-
-
-(** Could introduce a LOCALE with the assumptions
-     equiv A r;  congruent r b
-**)
+(* legacy ML bindings *)

-(*Conversion rule*)
-Goal "[| equiv A r;  congruent r b;  a: A |] \
-\     ==> (UN x:r{a}. b(x)) = b(a)";
-by (rtac (equiv_class_self RS UN_constant_eq) 1 THEN REPEAT (assume_tac 1));
-by (rewrite_goals_tac [equiv_def,congruent_def,sym_def]);
-by (blast_tac (claset() delrules [equalityI]) 1);
-qed "UN_equiv_class";
-
-(*type checking of  UN x:r{a}. b(x) *)
-val prems = Goalw [quotient_def]
-    "[| equiv A r;  congruent r b;  X: A//r;     \
-\       !!x.  x : A ==> b(x) : B |]             \
-\    ==> (UN x:X. b(x)) : B";
-by (cut_facts_tac prems 1);
-by (Clarify_tac 1);
-by (stac UN_equiv_class 1);
-by (REPEAT (ares_tac prems 1));
-qed "UN_equiv_class_type";
-
-(*Sufficient conditions for injectiveness.  Could weaken premises!
-  major premise could be an inclusion; bcong could be !!y. y:A ==> b(y):B
-*)
-val prems = Goalw [quotient_def]
-    "[| equiv A r;   congruent r b;  \
-\       (UN x:X. b(x))=(UN y:Y. b(y));  X: A//r;  Y: A//r;        \
-\       !!x y. [| x:A; y:A; b(x)=b(y) |] ==> (x,y):r |]         \
-\    ==> X=Y";
-by (cut_facts_tac prems 1);
-by (Clarify_tac 1);
-by (rtac equiv_class_eq 1);
-by (REPEAT (ares_tac prems 1));
-by (etac box_equals 1);
-by (REPEAT (ares_tac [UN_equiv_class] 1));
-qed "UN_equiv_class_inject";
-
-
-(**** Defining binary operations upon equivalence classes ****)
-
-
-Goalw [congruent_def,congruent2_def,equiv_def,refl_def]
-    "[| equiv A r;  congruent2 r b;  a: A |] ==> congruent r (b a)";
-by (Blast_tac 1);
-qed "congruent2_implies_congruent";
-
-Goalw [congruent_def]
-    "[| equiv A r;  congruent2 r b;  a: A |] ==> \
-\    congruent r (%x1. UN x2:r{a}. b x1 x2)";
-by (Clarify_tac 1);
-by (rtac (equiv_type RS subsetD RS SigmaE2) 1 THEN REPEAT (assume_tac 1));
-                                     congruent2_implies_congruent]) 1);
-by (rewrite_goals_tac [congruent2_def,equiv_def,refl_def]);
-by (blast_tac (claset() delrules [equalityI]) 1);
-qed "congruent2_implies_congruent_UN";
-
-Goal "[| equiv A r;  congruent2 r b;  a1: A;  a2: A |]  \
-\    ==> (UN x1:r{a1}. UN x2:r{a2}. b x1 x2) = b a1 a2";
-                                     congruent2_implies_congruent,
-                                     congruent2_implies_congruent_UN]) 1);
-qed "UN_equiv_class2";
-
-(*type checking*)
-val prems = Goalw [quotient_def]
-    "[| equiv A r;  congruent2 r b;  \
-\       X1: A//r;  X2: A//r;      \
-\       !!x1 x2.  [| x1: A; x2: A |] ==> b x1 x2 : B |]    \
-\    ==> (UN x1:X1. UN x2:X2. b x1 x2) : B";
-by (cut_facts_tac prems 1);
-by (Clarify_tac 1);
-by (REPEAT (ares_tac (prems@[UN_equiv_class_type,
-                             congruent2_implies_congruent_UN,
-                             congruent2_implies_congruent, quotientI]) 1));
-qed "UN_equiv_class_type2";
-
-(*Allows a natural expression of binary operators, without explicit calls
-  to "split"*)
-Goal "(UN (x1,x2):X. UN (y1,y2):Y. A x1 x2 y1 y2) = \
-\     (UN x:X. UN y:Y. (%(x1, x2). (%(y1, y2). A x1 x2 y1 y2) y) x)";
-by Auto_tac;
-qed "UN_UN_split_split_eq";
-
-(*Suggested by John Harrison -- the two subproofs may be MUCH simpler
-  than the direct proof*)
-val prems = Goalw [congruent2_def,equiv_def,refl_def]
-    "[| equiv A r;      \
-\       !! y z w. [| w: A;  (y,z) : r |] ==> b y w = b z w;      \
-\       !! y z w. [| w: A;  (y,z) : r |] ==> b w y = b w z       \
-\    |] ==> congruent2 r b";
-by (cut_facts_tac prems 1);
-by (Clarify_tac 1);
-by (blast_tac (claset() addIs (trans::prems)) 1);
-qed "congruent2I";
-
-val [equivA,commute,congt] = Goal
-    "[| equiv A r;      \
-\       !! y z. [| y: A;  z: A |] ==> b y z = b z y;        \
-\       !! y z w. [| w: A;  (y,z): r |] ==> b w y = b w z       \
-\    |] ==> congruent2 r b";
-by (resolve_tac [equivA RS congruent2I] 1);
-by (rtac (commute RS trans) 1);
-by (rtac (commute RS trans RS sym) 3);
-by (rtac sym 5);
-by (REPEAT (ares_tac [congt] 1
-     ORELSE etac (equivA RS equiv_type RS subsetD RS SigmaE2) 1));
-qed "congruent2_commuteI";
-
-
-(*** Cardinality results suggested by Florian Kammueller ***)
-
-(*Recall that  equiv A r  implies  r <= A <*> A   (equiv_type) *)
-Goal "[| finite A; r <= A <*> A |] ==> finite (A//r)";
-by (rtac finite_subset 1);
-by (etac (finite_Pow_iff RS iffD2) 2);
-by (rewtac quotient_def);
-by (Blast_tac 1);
-qed "finite_quotient";
-
-Goalw [quotient_def]
-    "[| finite A;  r <= A <*> A;  X : A//r |] ==> finite X";
-by (rtac finite_subset 1);
-by (assume_tac 2);
-by (Blast_tac 1);
-qed "finite_equiv_class";
-
-Goal "[| finite A;  equiv A r;  ALL X : A//r. k dvd card(X) |] \
-\     ==> k dvd card(A)";
-by (rtac (Union_quotient RS subst) 1);
-by (assume_tac 1);
-by (rtac dvd_partition 1);
-by (blast_tac (claset() addDs [quotient_disj]) 4);
-by (ALLGOALS
-    (asm_simp_tac (simpset() addsimps [Union_quotient, equiv_type,
-				       finite_quotient])));
-qed "equiv_imp_dvd_card";
+val UN_UN_split_split_eq = thm "UN_UN_split_split_eq";
+val UN_constant_eq = thm "UN_constant_eq";
+val UN_equiv_class = thm "UN_equiv_class";
+val UN_equiv_class2 = thm "UN_equiv_class2";
+val UN_equiv_class_inject = thm "UN_equiv_class_inject";
+val UN_equiv_class_type = thm "UN_equiv_class_type";
+val UN_equiv_class_type2 = thm "UN_equiv_class_type2";
+val Union_quotient = thm "Union_quotient";
+val comp_equivI = thm "comp_equivI";
+val congruent2I = thm "congruent2I";
+val congruent2_commuteI = thm "congruent2_commuteI";
+val congruent2_def = thm "congruent2_def";
+val congruent2_implies_congruent = thm "congruent2_implies_congruent";
+val congruent2_implies_congruent_UN = thm "congruent2_implies_congruent_UN";
+val congruent_def = thm "congruent_def";
+val eq_equiv_class = thm "eq_equiv_class";
+val eq_equiv_class_iff = thm "eq_equiv_class_iff";
+val equiv_class_eq = thm "equiv_class_eq";
+val equiv_class_eq_iff = thm "equiv_class_eq_iff";
+val equiv_class_nondisjoint = thm "equiv_class_nondisjoint";
+val equiv_class_self = thm "equiv_class_self";
+val equiv_class_subset = thm "equiv_class_subset";
+val equiv_comp_eq = thm "equiv_comp_eq";
+val equiv_def = thm "equiv_def";
+val equiv_imp_dvd_card = thm "equiv_imp_dvd_card";
+val equiv_type = thm "equiv_type";
+val finite_equiv_class = thm "finite_equiv_class";
+val finite_quotient = thm "finite_quotient";
+val quotientE = thm "quotientE";
+val quotientI = thm "quotientI";
+val quotient_def = thm "quotient_def";
+val quotient_disj = thm "quotient_disj";
+val refl_comp_subset = thm "refl_comp_subset";
+val subset_equiv_class = thm "subset_equiv_class";
+val sym_trans_comp_subset = thm "sym_trans_comp_subset";
--- a/src/HOL/Integ/Equiv.thy	Thu Aug 08 23:46:51 2002 +0200
+++ b/src/HOL/Integ/Equiv.thy	Thu Aug 08 23:47:41 2002 +0200
@@ -1,23 +1,273 @@
-(*  Title:      Equiv.thy
+(*  Title:      HOL/Integ/Equiv.thy
ID:         $Id$
Authors:    Lawrence C Paulson, Cambridge University Computer Laboratory
-
-Equivalence relations in Higher-Order Set Theory
*)

-Equiv = Relation + Finite_Set +
+header {* Equivalence relations in Higher-Order Set Theory *}
+
+theory Equiv = Relation + Finite_Set:
+
+subsection {* Equiv relations *}
+
+locale equiv =
+  fixes A and r
+  assumes refl: "refl A r"
+    and sym: "sym r"
+    and trans: "trans r"
+
+text {*
+  Suppes, Theorem 70: @{text r} is an equiv relation iff @{text "r\<inverse> O
+  r = r"}.
+
+  First half: @{text "equiv A r ==> r\<inverse> O r = r"}.
+*}
+
+lemma sym_trans_comp_subset:
+    "sym r ==> trans r ==> r\<inverse> O r \<subseteq> r"
+  by (unfold trans_def sym_def converse_def) blast
+
+lemma refl_comp_subset: "refl A r ==> r \<subseteq> r\<inverse> O r"
+  by (unfold refl_def) blast
+
+lemma equiv_comp_eq: "equiv A r ==> r\<inverse> O r = r"
+  apply (unfold equiv_def)
+  apply clarify
+  apply (rule equalityI)
+   apply (rules intro: sym_trans_comp_subset refl_comp_subset)+
+  done
+
+text {*
+  Second half.
+*}
+
+lemma comp_equivI:
+    "r\<inverse> O r = r ==> Domain r = A ==> equiv A r"
+  apply (unfold equiv_def refl_def sym_def trans_def)
+  apply (erule equalityE)
+  apply (subgoal_tac "\<forall>x y. (x, y) \<in> r --> (y, x) \<in> r")
+   apply fast
+  apply fast
+  done
+
+
+subsection {* Equivalence classes *}
+
+lemma equiv_class_subset:
+  "equiv A r ==> (a, b) \<in> r ==> r{a} \<subseteq> r{b}"
+  -- {* lemma for the next result *}
+  by (unfold equiv_def trans_def sym_def) blast
+
+lemma equiv_class_eq: "equiv A r ==> (a, b) \<in> r ==> r{a} = r{b}"
+  apply (assumption | rule equalityI equiv_class_subset)+
+  apply (unfold equiv_def sym_def)
+  apply blast
+  done
+
+lemma equiv_class_self: "equiv A r ==> a \<in> A ==> a \<in> r{a}"
+  by (unfold equiv_def refl_def) blast
+
+lemma subset_equiv_class:
+    "equiv A r ==> r{b} \<subseteq> r{a} ==> b \<in> A ==> (a,b) \<in> r"
+  -- {* lemma for the next result *}
+  by (unfold equiv_def refl_def) blast
+
+lemma eq_equiv_class:
+    "r{a} = r{b} ==> equiv A r ==> b \<in> A ==> (a, b) \<in> r"
+  by (rules intro: equalityD2 subset_equiv_class)
+
+lemma equiv_class_nondisjoint:
+    "equiv A r ==> x \<in> (r{a} \<inter> r{b}) ==> (a, b) \<in> r"
+  by (unfold equiv_def trans_def sym_def) blast
+
+lemma equiv_type: "equiv A r ==> r \<subseteq> A \<times> A"
+  by (unfold equiv_def refl_def) blast
+
+lemma equiv_class_eq_iff:
+  "equiv A r ==> ((x, y) \<in> r) = (r{x} = r{y} & x \<in> A & y \<in> A)"
+  by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
+
+lemma eq_equiv_class_iff:
+  "equiv A r ==> x \<in> A ==> y \<in> A ==> (r{x} = r{y}) = ((x, y) \<in> r)"
+  by (blast intro!: equiv_class_eq dest: eq_equiv_class equiv_type)
+
+
+subsection {* Quotients *}
+
constdefs
-  equiv    :: "['a set, ('a*'a) set] => bool"
-    "equiv A r == refl A r & sym(r) & trans(r)"
+  quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90)
+  "A//r == \<Union>x \<in> A. {r{x}}"  -- {* set of equiv classes *}
+
+lemma quotientI: "x \<in> A ==> r{x} \<in> A//r"
+  by (unfold quotient_def) blast
+
+lemma quotientE:
+  "X \<in> A//r ==> (!!x. X = r{x} ==> x \<in> A ==> P) ==> P"
+  by (unfold quotient_def) blast
+
+lemma Union_quotient: "equiv A r ==> Union (A//r) = A"
+  by (unfold equiv_def refl_def quotient_def) blast

-  quotient :: "['a set, ('a*'a) set] => 'a set set"  (infixl "'/'/" 90)
-    "A//r == UN x:A. {r{x}}"      (*set of equiv classes*)
+lemma quotient_disj:
+  "equiv A r ==> X \<in> A//r ==> Y \<in> A//r ==> X = Y | (X \<inter> Y = {})"
+  apply (unfold quotient_def)
+  apply clarify
+  apply (rule equiv_class_eq)
+   apply assumption
+  apply (unfold equiv_def trans_def sym_def)
+  apply blast
+  done
+
+
+subsection {* Defining unary operations upon equivalence classes *}
+
+locale congruent =
+  fixes r and b
+  assumes congruent: "(y, z) \<in> r ==> b y = b z"
+
+lemma UN_constant_eq: "a \<in> A ==> \<forall>y \<in> A. b y = c ==> (\<Union>y \<in> A. b(y))=c"
+  -- {* lemma required to prove @{text UN_equiv_class} *}
+  by auto
+
+lemma UN_equiv_class:
+  "equiv A r ==> congruent r b ==> a \<in> A
+    ==> (\<Union>x \<in> r{a}. b x) = b a"
+  -- {* Conversion rule *}
+  apply (rule equiv_class_self [THEN UN_constant_eq], assumption+)
+  apply (unfold equiv_def congruent_def sym_def)
+  apply (blast del: equalityI)
+  done

-  congruent  :: "[('a*'a) set, 'a=>'b] => bool"
-    "congruent r b  == ALL y z. (y,z):r --> b(y)=b(z)"
+lemma UN_equiv_class_type:
+  "equiv A r ==> congruent r b ==> X \<in> A//r ==>
+    (!!x. x \<in> A ==> b x : B) ==> (\<Union>x \<in> X. b x) : B"
+  apply (unfold quotient_def)
+  apply clarify
+  apply (subst UN_equiv_class)
+     apply auto
+  done
+
+text {*
+  Sufficient conditions for injectiveness.  Could weaken premises!
+  major premise could be an inclusion; bcong could be @{text "!!y. y \<in>
+  A ==> b y \<in> B"}.
+*}
+
+lemma UN_equiv_class_inject:
+  "equiv A r ==> congruent r b ==>
+    (\<Union>x \<in> X. b x) = (\<Union>y \<in> Y. b y) ==> X \<in> A//r ==> Y \<in> A//r
+    ==> (!!x y. x \<in> A ==> y \<in> A ==> b x = b y ==> (x, y) \<in> r)
+    ==> X = Y"
+  apply (unfold quotient_def)
+  apply clarify
+  apply (rule equiv_class_eq)
+   apply assumption
+  apply (subgoal_tac "b x = b xa")
+   apply blast
+  apply (erule box_equals)
+   apply (assumption | rule UN_equiv_class)+
+  done
+
+
+subsection {* Defining binary operations upon equivalence classes *}
+
+locale congruent2 =
+  fixes r and b
+  assumes congruent2:
+    "(y1, z1) \<in> r ==> (y2, z2) \<in> r ==> b y1 y2 = b z1 z2"
+
+lemma congruent2_implies_congruent:
+    "equiv A r ==> congruent2 r b ==> a \<in> A ==> congruent r (b a)"
+  by (unfold congruent_def congruent2_def equiv_def refl_def) blast
+
+lemma congruent2_implies_congruent_UN:
+  "equiv A r ==> congruent2 r b ==> a \<in> A ==>
+    congruent r (\<lambda>x1. \<Union>x2 \<in> r{a}. b x1 x2)"
+  apply (unfold congruent_def)
+  apply clarify
+  apply (rule equiv_type [THEN subsetD, THEN SigmaE2], assumption+)
+  apply (simp add: UN_equiv_class congruent2_implies_congruent)
+  apply (unfold congruent2_def equiv_def refl_def)
+  apply (blast del: equalityI)
+  done
+
+lemma UN_equiv_class2:
+  "equiv A r ==> congruent2 r b ==> a1 \<in> A ==> a2 \<in> A
+    ==> (\<Union>x1 \<in> r{a1}. \<Union>x2 \<in> r{a2}. b x1 x2) = b a1 a2"
+  by (simp add: UN_equiv_class congruent2_implies_congruent
+    congruent2_implies_congruent_UN)

-  congruent2 :: "[('a*'a) set, ['a,'a]=>'b] => bool"
-    "congruent2 r b == ALL y1 z1 y2 z2.
-                         (y1,z1):r --> (y2,z2):r --> b y1 y2 = b z1 z2"
+lemma UN_equiv_class_type2:
+  "equiv A r ==> congruent2 r b ==> X1 \<in> A//r ==> X2 \<in> A//r
+    ==> (!!x1 x2. x1 \<in> A ==> x2 \<in> A ==> b x1 x2 \<in> B)
+    ==> (\<Union>x1 \<in> X1. \<Union>x2 \<in> X2. b x1 x2) \<in> B"
+  apply (unfold quotient_def)
+  apply clarify
+  apply (blast intro: UN_equiv_class_type congruent2_implies_congruent_UN
+    congruent2_implies_congruent quotientI)
+  done
+
+lemma UN_UN_split_split_eq:
+  "(\<Union>(x1, x2) \<in> X. \<Union>(y1, y2) \<in> Y. A x1 x2 y1 y2) =
+    (\<Union>x \<in> X. \<Union>y \<in> Y. (\<lambda>(x1, x2). (\<lambda>(y1, y2). A x1 x2 y1 y2) y) x)"
+  -- {* Allows a natural expression of binary operators, *}
+  -- {* without explicit calls to @{text split} *}
+  by auto
+
+lemma congruent2I:
+  "equiv A r
+    ==> (!!y z w. w \<in> A ==> (y, z) \<in> r ==> b y w = b z w)
+    ==> (!!y z w. w \<in> A ==> (y, z) \<in> r ==> b w y = b w z)
+    ==> congruent2 r b"
+  -- {* Suggested by John Harrison -- the two subproofs may be *}
+  -- {* \emph{much} simpler than the direct proof. *}
+  apply (unfold congruent2_def equiv_def refl_def)
+  apply clarify
+  apply (blast intro: trans)
+  done
+
+lemma congruent2_commuteI:
+  assumes equivA: "equiv A r"
+    and commute: "!!y z. y \<in> A ==> z \<in> A ==> b y z = b z y"
+    and congt: "!!y z w. w \<in> A ==> (y, z) \<in> r ==> b w y = b w z"
+  shows "congruent2 r b"
+  apply (rule equivA [THEN congruent2I])
+   apply (rule commute [THEN trans])
+     apply (rule_tac [3] commute [THEN trans, symmetric])
+       apply (rule_tac [5] sym)
+       apply (assumption | rule congt |
+         erule equivA [THEN equiv_type, THEN subsetD, THEN SigmaE2])+
+  done
+
+
+subsection {* Cardinality results *}
+
+text {* (suggested by Florian Kammüller) *}
+
+lemma finite_quotient: "finite A ==> r \<subseteq> A \<times> A ==> finite (A//r)"
+  -- {* recall @{thm equiv_type} *}
+  apply (rule finite_subset)
+   apply (erule_tac [2] finite_Pow_iff [THEN iffD2])
+  apply (unfold quotient_def)
+  apply blast
+  done
+
+lemma finite_equiv_class:
+  "finite A ==> r \<subseteq> A \<times> A ==> X \<in> A//r ==> finite X"
+  apply (unfold quotient_def)
+  apply (rule finite_subset)
+   prefer 2 apply assumption
+  apply blast
+  done
+
+lemma equiv_imp_dvd_card:
+  "finite A ==> equiv A r ==> \<forall>X \<in> A//r. k dvd card X
+    ==> k dvd card A"
+  apply (rule Union_quotient [THEN subst])
+   apply assumption
+  apply (rule dvd_partition)
+     prefer 4 apply (blast dest: quotient_disj)
+    apply (simp_all add: Union_quotient equiv_type finite_quotient)
+  done
+
end`